Question

Consider an economy where the representative consumer has a utility function u (C; L) over consumption C and leisure L. Assume preferences satisfy the standard properties we saw in class. The consumer has an endowment of H units of time that they allocate to leisure or labor. The consumer also receives dividends, D, from the representative Örm. The representative consumer provides labor, Ns, at wage rate w, and receives dividends D, from the representative Örm. The representative Örm has a production technology given by Y = AF (K; N) where K is the Öxed capital input and N is labor input.

(a) (2 points) Set up the consumerís problem.

(b) (2 points) Solve the consumerís problem.in order to Önd a set of equations that implicitly allow you to Önd optimal labor supply N , optimal consumption, C , and optimal leisure, L , as a function of (w; D). Assume for the remaining parts of this exercise that the utility function is given by u(C; L) = log C (H L) 1+ 1 + ! over consumption C and leisure L; where ; > 0.

(c) (3 points) Use this speciÖcation for the utility function to Önd optimal labor supply N , optimal consumption, C , and optimal leisure, L , as a function of (w; D) from the optimality conditions you derived previously.

(d) (2 points) Use the above optimality conditions to assess how the optimal supply of labor, Ns , responds to a change in dividends.

(e) (2 points)How does this di§er from the example we saw in class?

(f) (2 points) How does optimal consumption respond to a change in dividends? What is going on?

(g) (2 points) Set up the problem of a representative Örmís problem that produces and sells goods.

(h) (2 points) Solve the representative Örmís problem.in order to derive an equation that implicitly allows you to Önd optimal labor demand N d , as a function of (w; K). Assume for the remaining parts of this exercise that the production function is AF(K; N) = AKN 1 ; 0 < < 1 and A > 0

(i) (2 points) Derive the optimal amount of labor that the Örm wants to hire as a function of capital, K, and the wage rate, w. What is the name of this function?

(j) (2 points) DeÖne (as we did in class) a competitive equilibrium for this economy. Competitive Equilibrium

(k) (3 points) Solve for the competitive equilibrium level of labor.

(l) (3 points) Use the above equilibrium conditions to assess how the equilibrium of labor, Ns , responds to a change in total factor productivity, A.

(m) (3 points)How does this di§er from the example we saw in class? Why?

Answer 1

(a) Each consumer treats as fixed and later maximises utility subject to his/her constraint. That is each consumer solves for,

In the set consumer problem above, the utility function is give to us in the question that is, and the as we assumed that the consumer assumes , the cost per unit of leisure or the best forgone amount of wage to be constant, Thus every consumer that steps in the market wants to maximise her/her utility as per their constraint.

(b) According to the question the function for the consumer is given over where is the wage rate and is the dividend and the production technology function is given as where is the fixed amount capital and is the labour supplied. since labour would either consumer the wage rate or enjoy over the dividend, we can set a Lagrange Multiplier,

As per the restrictions given on the utility function assure that there is a unique optimum which is characterized by the first order condition below,

Thus, we get

(c) On solving the Lagrange Equation for the question above we obtain an equation to for both and as

Thus we get the marginal rate of substitution of leisure is equal to the wage rate

Mathematically,

(d) Supply of labour is directly related to the wages given, that is as the wage rate increases the supply of labour increases and vice-versa. According to the question, the firm is alloting dividends as well to the labour. And as we can see that the firm receives as the rental rate of change in the capital. And the marginal rate of substitution of leisure is qeual to the wage rate, that is the consumers enoys leisure more as wage rate increases. Thus the nature of dividend, that is will be similar to that of the wage rate.